The evidence lower bound (ELBO), denoted as $\mathcal{L}$, is defined as: $\mathcal{L}(v,\theta,q)=\log p(v;\theta)-D_{KL} (q(h|v)||p(h|v;\theta))$, where $q$ is an arbitrary probability distribution over latent variables $h$, and $v$ represents observed variables. $\mathcal{L}$ always has at most the same value as the desired log-probability, since the difference between $\log p(v)$ and $\mathcal{L}(v,\theta,q)$ is given by the Kullback-Leibler (KL) divergence, which is always nonnegative. The two values are equal if and only if $q$ is the same distribution as the true posterior $p(h|v)$. $\mathcal{L}$ can be rearranged through algebra into the simpler form $\mathcal{L}(v,\theta,q)=\mathbb{E}_{h\sim q}[\log p(h,v)]+H(q)$. Thus, we can think of inference as the procedure for finding the distribution $q$ that maximizes $\mathcal{L}$.